Mathematics 124: Number Theory
Harvard College/GSAS: 111533
Fall 2012-2013
Barry C. Mazur
Meeting Time: Tuesday, Thursday 10:00am - 11:30am
Exam Group: 12,13
Factorization and the primes; congruences; quadratic residues and reciprocity; continued fractions and approximations; Pell's equation; selected Diophantine equations; theory of integral quadratic forms.
Prerequisite: Mathematics 122 (which may be taken concurrently) or equivalent.
Syllabus
- Homework Set due Sept 13
- What is Math 124?
- Introduction, Factorization, and Unique Factorization
- Unique Factorization, non-Unique Factorization, Intro to Dirichlet series
- Dirichlet series, Arithmetic Functions, Dirichlet.Product
- Congruences, The Chinese Remainder Theorem, and Homework set due Sept 20
- Euler Phi-function and Congruence mod p
- Quadratic equations mod p; Wilson's Theorem and Homework set due Sept 27
- The structure of (Z/pZ)*
- Algebraic Numbers,Quadratic irrationalities, and Homework set due Oct 4
- More Gaussian Integers, Relationship between Pell's Equation, Units in Real Quadratic fields and Continued Fractions; AND Homework set due Oct 11
- Pell’s Equation, Units in real quadratic fields, Continued fractions, Approximations.
- More continued fractions and Homework set due October 18
- Continued fractions for Pell's Equation
- Even more continued fractions for Pell’s Equation. Introduction to Quadratic Reciprocity and Homework due October 25
- Quadratic Reciprocity
- Yet more quadratic reciprocity, Quadratic Gauss Sums, and Homework set due Nov 1
- Quadratic Gauss Sums again; Finite fields.
- More Gauss sums and intro to finite fields
- Finite Fields. Characters. Homework set due November 15
- Binary Quadratic Forms and the first part of the Homework set due Nov 27
- More binary quadratic forms; Homogenous Forms over finite fields, and both parts of Homework set due Nov 27
- Warning-Chevalley, and Lagrange's Theorem
- Homogenous Forms over Finite Fields; and back to Jaobi Sums